Pairwise orthogonal frames generated by regular representations of LCA groups

Abstract

Having potential applications in multiplexing techniques and in the synthesis of frames, orthogonality (or strongly disjointness) plays a significant role in frame theory (e.g. construction of new frames from existing ones, constructions related with duality, etc.). In this article, we study orthogonality of a pair of frames over locally compact abelian (LCA) groups. We start with the investigation of the dual Gramian analysis tools of Ron and Shen through a pre-Gramian operator over the set-up of LCA groups. Then we fiberize some operators associated with Bessel families generated by unitary actions of co-compact (not necessarily discrete) subgroups of LCA groups. Using this fiberization, we study and characterize a pair of orthogonal frames generated by the action of a unitary representation ρ of a co-compact subgroup Γ⊂G on a separable Hilbert space L2(G), where G is a second countable LCA group. Precisely, we consider frames of the form {ρ(γ)ψ:γ∈Γ,ψ∈Ψ} for a countable family Ψ in L2(G). We pay special attention to this problem in the context of translation-invariant space by assuming ρ as the action of Γ on L2(G) by left-translation. The representation of Γ acting on L2(G) by (left-)translation is called the (left-)regular representation of Γ. Further, we apply our results on co-compact Gabor systems over LCA groups. At this juncture, it is pertinent to note that the resulting characterization can be useful for constructing new frames by using various techniques including the unitary extension principle by Ron and Shen [24] and its recent extension to LCA groups by Christensen and Goh [7].